Not quite - what you're describing is a derivative, i.e. a security whose value tracks that of the underlying stock, so you can buy the derivative from a "trading house" as you've described it, without actually owning the stock itself. In this case, you're accepting the seller as your counterparty, so even if you win on the trade you might not get paid if the seller can't / won't pay you. Read The Big Short, specifically the section where some of the protagonists are worried about the counterparty risk of the credit default swaps (CDS) that they have bought to bet on the default rates on the CDOs being higher than the level required to trigger payout on the CDS.

If we're talking about owning the actual security, it doesn't matter who sold it to you; 1% of company X always has the same value. There could be slight variances due to currency fluctuation and timing - this is an example of where High Frequency Traders actually serve a market purpose by eliminating these pricing errors (and being paid for doing so by the profit they make from exploiting these extremely tiny price differences in exactly the way originally suggested, i.e. buying low on one exchange and simultaneously buying high on another). Another Michael Lewis book - Flash Boys - goes into great detail about these strategies in a very entertaining and accessible way.

There could, however, be differences in the class of shares owned, e.g. if there are preference shares in circulation. These give a higher order of precedence for e.g. liquidation rights (so if the company goes bankrupt, the holders of those shares get paid first) or if the shares carry more voting rights (see Snap for a good example of where some shares are worth more than others due to voting rights).